{
 "cells": [
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   "cell_type": "markdown",
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   },
   "source": [
    "# Random Signals\n",
    "\n",
    "*This jupyter notebook is part of a [collection of notebooks](../index.ipynb) on various topics of Digital Signal Processing. Please direct questions and suggestions to [Sascha.Spors@uni-rostock.de](mailto:Sascha.Spors@uni-rostock.de).*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Auto Power Spectral Density\n",
    "\n",
    "The (auto-) [power spectral density](https://en.wikipedia.org/wiki/Spectral_density#Power_spectral_density) (PSD) is defined as the Fourier transformation of the [auto-correlation function](correlation_functions.ipynb) (ACF)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Definition\n",
    "\n",
    "For a continuous-amplitude, real-valued, wide-sense stationary (WSS) random signal $x[k]$ the PSD is given as\n",
    "\n",
    "\\begin{equation}\n",
    "\\Phi_{xx}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega}) = \\mathcal{F}_* \\{ \\varphi_{xx}[\\kappa] \\},\n",
    "\\end{equation}\n",
    "\n",
    "where $\\mathcal{F}_* \\{ \\cdot \\}$ denotes the [discrete-time Fourier transformation](https://en.wikipedia.org/wiki/Discrete-time_Fourier_transform) (DTFT) and $\\varphi_{xx}[\\kappa]$ the ACF of $x[k]$. Note that the DTFT is performed with respect to $\\kappa$. The ACF of a random signal of finite length $N$ can be expressed by way of a linear convolution\n",
    "\n",
    "\\begin{equation}\n",
    "\\varphi_{xx}[\\kappa] = \\frac{1}{N} \\cdot x_N[k] * x_N[-k].\n",
    "\\end{equation}\n",
    "\n",
    "Taking the DTFT of the left- and right-hand side results in\n",
    "\n",
    "\\begin{equation}\n",
    "\\Phi_{xx}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega}) = \\frac{1}{N} \\, X_N(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega})\\, X_N(\\mathrm{e}^{-\\,\\mathrm{j}\\,\\Omega}) = \n",
    "\\frac{1}{N} \\, | X_N(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega}) |^2.\n",
    "\\end{equation}\n",
    "\n",
    "The last equality results from the definition of the magnitude and the symmetry of the DTFT for real-valued signals. The spectrum $X_N(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega})$ quantifies the amplitude density of the signal $x_N[k]$. It can be concluded from above result that the PSD quantifies the squared amplitude or power density of a random signal. This explains the term power spectral density."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Properties\n",
    "\n",
    "The properties of the PSD can be deduced from the properties of the ACF and the DTFT as:\n",
    "\n",
    "1. From the link between the PSD $\\Phi_{xx}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega})$ and the spectrum $X_N(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega})$ derived above it can be concluded that the PSD is real valued\n",
    "\n",
    "    $$\\Phi_{xx}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega}) \\in \\mathbb{R}$$\n",
    "\n",
    "2. From the even symmetry $\\varphi_{xx}[\\kappa] = \\varphi_{xx}[-\\kappa]$ of the ACF it follows that\n",
    "\n",
    "    $$ \\Phi_{xx}(\\mathrm{e}^{\\,\\mathrm{j} \\, \\Omega}) = \\Phi_{xx}(\\mathrm{e}^{\\,-\\mathrm{j}\\, \\Omega}) $$\n",
    "\n",
    "3. The PSD of an uncorrelated random signal is given as\n",
    "\n",
    "    $$ \\Phi_{xx}(\\mathrm{e}^{\\,\\mathrm{j} \\, \\Omega}) = (\\sigma_x^2 + \\mu_x^2) \\cdot {\\bot \\!\\! \\bot \\!\\! \\bot}\\left( \\frac{\\Omega}{2 \\pi} \\right) ,$$\n",
    "    \n",
    "    which can be deduced from the [ACF of an uncorrelated signal](correlation_functions.ipynb#Properties).\n",
    "\n",
    "4. The quadratic mean of a random signal is given as\n",
    "\n",
    "    $$ E\\{ x[k]^2 \\} = \\varphi_{xx}[\\kappa=0] = \\frac{1}{2\\pi} \\int\\limits_{-\\pi}^{\\pi} \\Phi_{xx}(\\mathrm{e}^{\\,\\mathrm{j}\\, \\Omega}) \\,\\mathrm{d} \\Omega $$\n",
    "\n",
    "    The last relation can be found by expressing the ACF via the inverse DTFT of $\\Phi_{xx}$ and considering that $\\mathrm{e}^{\\mathrm{j} \\Omega \\kappa} = 1$ when evaluating the integral for $\\kappa=0$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example - Power Spectral Density of a Speech Signal\n",
    "\n",
    "In this example the PSD $\\Phi_{xx}(\\mathrm{e}^{\\,\\mathrm{j} \\,\\Omega})$ of a speech signal of length $N$ is estimated by applying a discrete Fourier transformation (DFT) to its ACF. For a better interpretation of the PSD, the frequency axis $f = \\frac{\\Omega}{2 \\pi} \\cdot f_s$ has been chosen for illustration, where $f_s$ denotes the sampling frequency of the signal. The speech signal constitutes a recording of the vowel 'o' spoken from a German male, loaded into variable `x`.\n",
    "\n",
    "In Python the ACF is stored in a vector with indices $0, 1, \\dots, 2N - 2$ corresponding to the lags $\\kappa = (0, 1, \\dots, 2N - 2)^\\mathrm{T} - (N-1)$. When computing the discrete Fourier transform (DFT) of the ACF numerically by the fast Fourier transform (FFT) one has to take this shift into account. For instance, by multiplying the DFT $\\Phi_{xx}[\\mu]$ by $\\mathrm{e}^{\\mathrm{j} \\mu \\frac{2 \\pi}{2N - 1} (N-1)}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
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      "text/plain": [
       "<Figure size 960x384 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy.io import wavfile\n",
    "%matplotlib inline\n",
    "\n",
    "# read audio file\n",
    "fs, x = wavfile.read('../data/vocal_o_8k.wav')\n",
    "x = np.asarray(x, dtype=float)\n",
    "N = len(x)\n",
    "\n",
    "# compute ACF\n",
    "acf = 1/N * np.correlate(x, x, mode='full')\n",
    "# compute PSD\n",
    "psd = np.fft.fft(acf)\n",
    "psd = psd * np.exp(1j*np.arange(2*N-1)*2*np.pi*(N-1)/(2*N-1))\n",
    "f = np.fft.fftfreq(2*N-1, d=1/fs)\n",
    "\n",
    "# plot PSD\n",
    "plt.figure(figsize = (10, 4))\n",
    "plt.plot(f, np.real(psd))\n",
    "plt.title('Estimated power spectral density')\n",
    "plt.ylabel(r'$\\hat{\\Phi}_{xx}(e^{j \\Omega})$')\n",
    "plt.xlabel(r'$f / Hz$')\n",
    "plt.axis([0, 500, 0, 1.1*max(np.abs(psd))])\n",
    "plt.grid()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise**\n",
    "\n",
    "* What does the PSD tell you about the average spectral contents of a speech signal?\n",
    "\n",
    "Solution: The speech signal exhibits a harmonic structure with the dominant fundamental frequency $f_0 \\approx 100$ Hz and a number of harmonics $f_n \\approx n \\cdot f_0$ for $n > 0$. This due to the fact that vowels generate random signals which are in good approximation periodic. To generate vowels, the sound produced by the periodically vibrating vowel folds is filtered by the resonance volumes and articulators above the voice box. The spectrum of periodic signals is a line spectrum."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Cross-Power Spectral Density\n",
    "\n",
    "The cross-power spectral density is defined as the Fourier transformation of the [cross-correlation function](correlation_functions.ipynb#Cross-Correlation-Function) (CCF)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Definition\n",
    "\n",
    "For two continuous-amplitude, real-valued, wide-sense stationary (WSS) random signals $x[k]$ and $y[k]$, the cross-power spectral density is given as\n",
    "\n",
    "\\begin{equation}\n",
    "\\Phi_{xy}(\\mathrm{e}^{\\,\\mathrm{j} \\, \\Omega}) = \\mathcal{F}_* \\{ \\varphi_{xy}[\\kappa] \\},\n",
    "\\end{equation}\n",
    "\n",
    "where $\\varphi_{xy}[\\kappa]$ denotes the CCF of $x[k]$ and $y[k]$. Note again, that the DTFT is performed with respect to $\\kappa$. The CCF of two random signals of finite length $N$ and $M$ can be expressed by way of a linear convolution\n",
    "\n",
    "\\begin{equation}\n",
    "\\varphi_{xy}[\\kappa] = \\frac{1}{N} \\cdot x_N[k] * y_M[-k].\n",
    "\\end{equation}\n",
    "\n",
    "Note the chosen $\\frac{1}{N}$-averaging convention corresponds to the length of signal $x$. If $N \\neq M$, care should be taken on the interpretation of this normalization. In case of $N=M$ the $\\frac{1}{N}$-averaging yields a [biased estimator](https://en.wikipedia.org/wiki/Bias_of_an_estimator) of the CCF, which consistently should be denoted with $\\hat{\\varphi}_{xy,\\mathrm{biased}}[\\kappa]$.\n",
    "\n",
    "\n",
    "Taking the DTFT of the left- and right-hand side from above cross-correlation results in\n",
    "\n",
    "\\begin{equation}\n",
    "\\Phi_{xy}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega}) = \\frac{1}{N} \\, X_N(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega})\\, Y_M(\\mathrm{e}^{-\\,\\mathrm{j}\\,\\Omega}).\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Properties\n",
    "\n",
    "1. The symmetries of $\\Phi_{xy}(\\mathrm{e}^{\\,\\mathrm{j}\\, \\Omega})$ can be derived from the symmetries of the CCF and the DTFT as\n",
    "\n",
    "    $$ \\underbrace {\\Phi_{xy}(\\mathrm{e}^{\\,\\mathrm{j}\\, \\Omega}) = \\Phi_{xy}^*(\\mathrm{e}^{-\\,\\mathrm{j}\\, \\Omega})}_{\\varphi_{xy}[\\kappa] \\in \\mathbb{R}} = \n",
    "\\underbrace {\\Phi_{yx}(\\mathrm{e}^{\\,- \\mathrm{j}\\, \\Omega}) = \\Phi_{yx}^*(\\mathrm{e}^{\\,\\mathrm{j}\\, \\Omega})}_{\\varphi_{yx}[-\\kappa] \\in \\mathbb{R}},$$\n",
    "\n",
    "    from which $|\\Phi_{xy}(\\mathrm{e}^{\\,\\mathrm{j}\\, \\Omega})| = |\\Phi_{yx}(\\mathrm{e}^{\\,\\mathrm{j}\\, \\Omega})|$ can be concluded.\n",
    "\n",
    "2. The cross PSD of two uncorrelated random signals is given as\n",
    "\n",
    "    $$ \\Phi_{xy}(\\mathrm{e}^{\\,\\mathrm{j} \\, \\Omega}) = \\mu_x^2 \\mu_y^2 \\cdot {\\bot \\!\\! \\bot \\!\\! \\bot}\\left( \\frac{\\Omega}{2 \\pi} \\right) $$\n",
    "    \n",
    "    which can be deduced from the CCF of an uncorrelated signal."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example - Cross-Power Spectral Density\n",
    "\n",
    "The following example estimates and plots the cross PSD $\\Phi_{xy}(\\mathrm{e}^{\\,\\mathrm{j}\\, \\Omega})$ of two random signals $x_N[k]$ and $y_M[k]$ of finite lengths $N = 64$ and $M = 512$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
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       "<Figure size 960x384 with 1 Axes>"
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   "source": [
    "N = 64  # length of x\n",
    "M = 512  # length of y\n",
    "\n",
    "# generate two uncorrelated random signals\n",
    "np.random.seed(1)\n",
    "x = 2 + np.random.normal(size=N)\n",
    "y = 3 + np.random.normal(size=M)\n",
    "N = len(x)\n",
    "M = len(y)\n",
    "\n",
    "# compute cross PSD via CCF\n",
    "acf = 1/N * np.correlate(x, y, mode='full')\n",
    "psd = np.fft.fft(acf)\n",
    "psd = psd * np.exp(1j*np.arange(N+M-1)*2*np.pi*(M-1)/(2*M-1))\n",
    "psd = np.fft.fftshift(psd)\n",
    "Om = 2*np.pi * np.arange(0, N+M-1) / (N+M-1)\n",
    "Om = Om - np.pi\n",
    "\n",
    "# plot results\n",
    "plt.figure(figsize=(10, 4))\n",
    "plt.stem(Om, np.abs(psd), basefmt='C0:', use_line_collection=True)\n",
    "plt.title('Biased estimator of cross power spectral density')\n",
    "plt.ylabel(r'$|\\hat{\\Phi}_{xy}(e^{j \\Omega})|$')\n",
    "plt.xlabel(r'$\\Omega$')\n",
    "plt.grid()"
   ]
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   "source": [
    "**Exercise**\n",
    "\n",
    "* What does the cross PSD $\\Phi_{xy}(\\mathrm{e}^{\\,\\mathrm{j} \\, \\Omega})$ tell you about the statistical properties of the two random signals?\n",
    "\n",
    "Solution: The cross PSD $\\Phi_{xy}(\\mathrm{e}^{\\,\\mathrm{j} \\, \\Omega})$ is essential only non-zero for $\\Omega=0$. It hence can be concluded that the two random signals are not mean-free and uncorrelated to each other."
   ]
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   "source": [
    "**Copyright**\n",
    "\n",
    "This notebook is provided as [Open Educational Resource](https://en.wikipedia.org/wiki/Open_educational_resources). Feel free to use the notebook for your own purposes. The text is licensed under [Creative Commons Attribution 4.0](https://creativecommons.org/licenses/by/4.0/), the code of the IPython examples under the [MIT license](https://opensource.org/licenses/MIT). Please attribute the work as follows: *Sascha Spors, Digital Signal Processing - Lecture notes featuring computational examples*."
   ]
  }
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